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Problems 97
% check
>> error = max(abs(xa - exp(-1000*abs(t))))
error = 0.1679
The maximum error in this case is 0.1679, which is significant and cannot be
attributed to the nonideal interpolation or nonband-limitedness of x a(t). From
the bottom plot in Figure 3.19 observe that the reconstructed signal again
differs from the actual one in many places over the interpolated regions. □
From these examples it is clear that for practical purposes the spline
interpolation provides the best results.
3.5 PROBLEMS
P3.1 Using the matrix-vector multiplication approach discussed in this chapter, write a
MATLAB function to compute the DTFT of a finite-duration sequence. The format of
the function should be
function [X] = dtft(x,n,w)
% Computes Discrete-time Fourier Transform
% [X] = dtft(x,n,w)
% X = DTFT values computed at w frequencies
% x = finite duration sequence over n
% n = sample position vector
% w = frequency location vector
Use this function to compute the DTFT X(e jω )ofthe following finite-duration sequences
over −π ≤ ω ≤ π. Plot DTFT magnitude and angle graphs in one figure window.
1. x(n) =(0.6) |n| [u(n + 10) − u(n − 11)]. Comment on the angle plot.
2. x(n) =n(0.9) n [u(n) − u(n − 21)].
3. x(n) =[cos(0.5πn)+j sin(0.5πn)][u(n) − u(n − 51)]. Comment on the magnitude plot.
4. x(n) ={4, 3, 2, 1, 1, 2, 3, 4}. Comment on the angle plot.
↑
5. x(n) ={4, 3, 2, 1, −1, −2, −3, −4}. Comment on the angle plot.
↑
P3.2 Let x 1(n) ={1, 2, 2, 1}. Anew sequence x 2(n) isformed using
↑
{ x1(n), 0 ≤ n ≤ 3;
x 2(n) = x 1(n − 4), 4 ≤ n ≤ 7;
0, Otherwise.
1. Express X 2(e jω )interms of X 1(e jω ) without explicitly computing X 1(e jω ).
2. Verify your result using MATLAB by computing and plotting magnitudes of the
respective DTFTs.
(3.44)
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